TheOdd Cycle Transversal problem (oct) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of oct with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Od...

the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

Consider an undirected weighted graph G = (V,E) with |V | = n and |E| = m, where each vertex v ∈ V is assigned a label from a set of labels L = {λ1, ..., λl}. We show how to construct a compact distance oracle that can answer queries of the form: “what is the distance from v to the closest λ-labeled node” for a given node v ∈ V and label λ ∈ L. This problem was introduced by Hermelin, Levy, Wei...

• Introduce several new distance-based global and local functions based the smallest distance which are related to eccentricity, center, sum of eccentricities in graphs trees. The values acompared with some sharp bounds established. difference between eccentricity uniformity behaves a very similar way as itself. Motivated from study trees, we introduce on vertex leaf (called “uniformity” at tha...

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic ce...

A Diophantine figure, see i.e. [4, 5, 6], is a set of points on the integer grid Z where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in Z (the coordinates) and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diopha...

For a connected graph G = (V,E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u− v monophonic path in G. For subsets A and B of V , the monophonic distance dm(A,B) is defined as dm(A,B) = min{dm(x, y) : x ∈ A, y ∈ B}. A u− v path of length dm(A,B) is called an A−B detour monophonic path joining the sets A,B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called...

*Correspondence: [email protected] Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran Abstract In (Hamzeh et al. in Stud. Univ. Babeş-Bolyai, Chem., 4:73-85, 2012), we introduced a generalization of a degree distance of graphs as a new topological index. In this paper, we characterize the n-vertex tricyclic graphs whi...

Let G be a connected graph. The eccentricity of a vertex v is defined as the distance in G between v and a vertex farthest from v. The nondecreasing sequence of the eccentricities of the vertices of G is the eccentric sequence of G. In this paper, we characterize eccentric sequences of maximal outerplanar graphs.

The Fibonacci cube Γn is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. The eccentricity of a vertex u, denoted eG(u) is the greatest distance between u and any other vertex v in the graph G. For a given vertex u of Γn we characterize the vertices v such that d...